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3: Set Theory and Logic

Last updated

Oct 29, 2021
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- 2.4: Installments
- 3.1: The Basics

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Topic hierarchy

3.1: The Basics
The concept of sets is a fundamental idea in mathematics. Even at the earliest stage of mathematical reasoning, the idea of sets is being used. As such a fundamental concept, this subject can be studied on its own as a graduate-level course! Of course, we will not go into such detail as one would expect in such a graduate-level course. Here we will explore some basic definitions and relationships about sets so that we can ultimately apply them to some practical situations.
3.2: Operations with Sets
Commonly sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets. The union of two sets contains all the elements contained in either set (or both sets). The intersection of two sets contains only the elements that are in both sets. The complement of a set A contains everything that is not in the set A .
3.3: Venn Diagrams
To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called Venn Diagrams. A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets. Basic Venn diagrams can illustrate the interaction of two or three sets.
3.4: Inductive and Deductive Reasoning
A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments. An inductive argument uses a collection of specific examples as its premises and uses them to propose a general conclusion. A deductive argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.
3.5: Using Venn Diagrams to Analyze Arguments
3.6: Common Valid and Invalid Arguments
In the previous discussion, we saw that logical arguments can be invalid when the premises are not true, when the premises are not sufficient to guarantee the conclusion, or when there are invalid chains in logic. There are a number of other ways in which arguments can be invalid, a sampling of which are given here.
3.7: Truth Tables
3.8: Analyzing Arguments with Truth Tables

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